Artificial Viscosity Model to Mitigate Numerical Artefacts at Fluid Interfaces with Surface Tension

Computers and Fluids 143 (2017) 59–72

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Computers and Fluids

journal homepage: www.elsevier.com/locate/compfluid

Artiﬁcial viscosity model to mitigate numerical artefacts at ﬂuid interfaces with surface tension

∗ Fabian Denner a, , Fabien Evrard a, Ricardo Serfaty b, Berend G.M. van Wachem a a Thermoﬂuids Division, Department of Mechanical Engineering, Imperial College London, Exhibition Road, London, SW7 2AZ, United Kingdom b PetroBras, CENPES, Cidade Universitária, Avenida 1, Quadra 7, Sala 2118, Ilha do Fundão, Rio de Janeiro, Brazil

a r t i c l e i n f o a b s t r a c t

Article history: The numerical onset of parasitic and spurious artefacts in the vicinity of ﬂuid interfaces with surface ten- Received 21 April 2016 sion is an important and well-recognised problem with respect to the accuracy and numerical stability of

Revised 21 September 2016 interfacial flow simulations. Issues of particular interest are spurious capillary waves, which are spatially Accepted 14 November 2016 underresolved by the computational mesh yet impose very restrictive time-step requirements, as well as Available online 15 November 2016 parasitic currents, typically the result of a numerically unbalanced curvature evaluation. We present an Keywords: artificial viscosity model to mitigate numerical artefacts at surface-tension-dominated interfaces without Capillary waves adversely affecting the accuracy of the physical solution. The proposed methodology computes an addi- Parasitic currents tional interfacial shear stress term, including an interface viscosity, based on the local flow data and fluid Surface tension properties that reduces the impact of numerical artefacts and dissipates underresolved small scale inter-

Curvature face movements. Furthermore, the presented methodology can be readily applied to model surface shear Interfacial ﬂows viscosity, for instance to simulate the dissipative effect of surface-active substances adsorbed at the interface. The presented analysis of numerical test cases demonstrates the eﬃcacy of the proposed methodology in diminishing the adverse impact of parasitic and spurious interfacial artefacts on the convergence and stability of the numerical solution algorithm as well as on the overall accuracy of the simulation results. ©2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license. ( http://creativecommons.org/licenses/by/4.0/ )

1. Introduction currence in the modelling of surface-tension-dominated flows. Pre- vious studies [3–6] have identified two distinct origins of para- The dynamics of an interface separating two immiscible fluids sitic currents: a) a discrete imbalance between pressure “jump” is governed by the behaviour of individual molecules, which typi- and surface tension and b) a numerically unbalanced evaluation cally have a size of less than one nanometer. Computational fluid of the interface curvature. The imbalance of the pressure gradi- dynamics (CFD), however, is based on continuum mechanics and ent and surface tension can be eliminated by employing so-called treats fluids as continuous media. Molecular scales cannot be re- balanced-force discretisation methods that assure a discrete bal- solved using continuum mechanics, which leads to a variety of ance between surface tension and pressure gradient, which has theoretical and numerical difficulties with respect to the represen- previously been proposed and successfully demonstrated by Re- tation of fluid interfaces using CFD. These difficulties manifest as nardy and Renardy [14] and Francois et al. [4] for sharp surface parasitic (of unphysical origin) or spurious (of physical origin but representations such as the ghost-fluid method [15,16] , and by numerically misrepresented) flow features in the vicinity of the in- Francois et al. [4] , Mencinger and Žun [5] and Denner and van terface. Numerical artefacts of particular practical interest in inter- Wachem [6] for methods using a continuum surface force formu- facial flow modelling that have attracted significant attention and lation [1,17] . A numerically unbalanced evaluation of the interface research efforts are parasitic currents [1–7] and spurious capillary curvature, which is typically associated with spatial aliasing errors waves [1,8–13] . resulting from the computation of the second derivative of a dis- The numerical onset of unphysical flow currents in the vicin- crete indicator function or reconstruction [3,6] , leads to an unphys- ity of the interface, so-called parasitic currents , are a common oc- ical contribution to the momentum equations (via surface tension) and, consequently, causes an unphysical acceleration of the flow in the vicinity of the interface [4,7] . Using a balanced-force discretisa- ∗ Corresponding author: tion of the surface tension, parasitic currents are solely dependent E-mail address: [email protected] (F. Denner). http://dx.doi.org/10.1016/j.compfluid.2016.11.006 0045-7930/© 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license. ( http://creativecommons.org/licenses/by/4.0/ ) 60 F. Denner et al. / Computers and Fluids 143 (2017) 59–72 on the evaluation of the interface curvature and can, for instance, methodology for finite volume methods and reported that this be eliminated ( i.e. reduced to machine precision) for certain cases semi-implicit formulation of surface tension allows to exceed the by imposing the geometrically exact curvature [4–7] . Although pre- capillary time-step constraint by up to factor five without desta- vious studies [7,18] have shown that, with a careful discretisation bilising the solution of the presented two-dimensional test cases. of the interface curvature, parasitic currents can be reduced to ma- Schroeder et al. [11] proposed a two-dimensional method with a chine precision in some cases once the interface has reached a nu- semi-implicit implementation of surface tension on a Lagrangian merical equilibrium, parasitic currents are still an issue of signif- interface mesh ( i.e. using an explicit representation of the inter- icant practical relevance for applications with evolving interfaces, face) coupled to a Eulerian mesh for the flow, presenting sta- as evident by the considerable body of literature on this subject ble results for time-steps up to t = 3 t σ . In a similar fashion, published over the past five years alone ( e.g. [7,19–22] ). Zheng et al. [12] recently proposed a fully-implicit coupling of a Another important numerical artefact in interfacial flows are Lagrangian interface mesh to a MAC grid and showed that the spurious capillary waves . In a numerical framework, the shortest method can yield stable results for time-steps up to t ≈ 10 3 t σ . wavelength unambiguously resolved by the computational mesh is However, with respect to methods that rely on an implicit interface λ = , min 2 x with x representing the mesh spacing. Because an representation, such as Volume-of-Fluid methods [24] or Level-Set adequate spatial resolution of waves requires at least 6 − 10 cells methods [25,26] , and the CSF method of Brackbill et al. [1] , Denner λ ≤ per wavelength [13], capillary waves with a wavelength of min and van Wachem [13] demonstrated that the temporal resolution λ ࣠ λ 3 min are not part of the physical solution and can, therefore, requirements associated with the propagation of capillary waves be considered to be spurious capillary waves, meaning that these is a result of the spatiotemporal sampling of capillary waves and waves are a response of the discretised governing equations to a is independent of whether surface tension is implemented explicit perturbation of the interface but are numerically misrepresented. or implicit. Simulating the thermocapillary migration of a spheri- Therefore, spurious capillary waves are, contrary to parasitic cur- cal drop, Denner and van Wachem [13] demonstrated that with- rents, not the result of numerical errors but the result of the lim- out external perturbations acting at the interface (such as para- itations associated with the finite resolution of the computational sitic currents), the capillary time-step constraint can be exceeded mesh. The origin of spurious capillary waves can be physical per- by several orders of magnitude without destabilising the solution, turbations of the interface, for instance due to the collision of the presenting stable results for t = 10 4 t σ using an explicit imple- interface with an obstacle, as well as numerical perturbations, for mentation of surface tension. Thus, in order to mitigate or lift the instance the finite accuracy of numerical algorithms, discretisation capillary time-step constraint for numerical methods that rely on errors or parasitic currents [13] . An adequate temporal resolution an implicit interface representation, the shortest capillary waves of the propagation of all spatially resolved capillary waves is essen- spatially resolved by the computational mesh have to be either fil- tial for a stable numerical solution [1,13] . The dispersion relation of tered out or damped with an appropriate method to mitigate their capillary waves in inviscid fluids is given as [23] impact. Issues regarding numerical artefacts are, however, not limited σ 3 2 k ω σ = , (1) to interfacial flows. For instance, numerical oscillations induced by ρ + ρ a b a high-order discretisation of advection terms is a longstanding is- from which the phase velocity follows as c σ = ω σ /k, where ω σ is sue in CFD as well as numerical heat and mass transfer, and has the angular frequency of capillary waves, k is the wavenumber, σ been the topic of extensive studies, e.g. [27–31] . Artificial viscos- is the surface tension coefficient and ρ is the density of the ad- ity is a well-established concept to mitigate or eliminate high- jacent fluids a and b. Hence, the phase speed of capillary waves frequency oscillations in the solution and improve the stability of increases with decreasing wavelength. This anomalously dispersive the numerical methodology, in particular for shock capturing and behaviour of capillary waves leads to a very rigid time-step restric- in transsonic flows, and numerical models that incorporate arti- tion for interfacial flow simulations. For the shortest spatially re- ficial viscosity span a wide range of explicit and implicit meth- solved capillary waves, Denner and van Wachem [13] devised the ods, see e.g. [32–37] . Cook and Cabot [35] suggested that the ar- capillary time-step constraint as tificial grid-dependent viscosity should be chosen as to only damp wavenumbers close to the Nyquist wavenumber π/ x , which in a x t σ ≤ √ , (2) numerical simulation is the wavenumber of the shortest spatially + 2 cσ u resolved waves. Discretisation schemes which introduce numeri- where x denotes the mesh spacing and u is the flow velocity cal diffusion to avoid oscillatory solutions, such as TVD schemes tangential to the interface. Denner and van Wachem [13] demon- [27–29] , are often considered to be part of the artificial viscosity strated that the shortest spatially resolved capillary waves are in models as well. In fact, TVD schemes can be readily translated into most cases not subject to viscous attenuation, since the computa- an explicit artificial viscosity term, as for instance shown by Davis tional mesh is usually too coarse for typical values of the fluid vis- [38] . cosities to spatially resolve the vorticity generated by the shortest In this study we propose an artificial viscosity model to miti- numerically represented waves. gate numerical artefacts at fluid interfaces, expanding on the work The capillary time-step constraint limits the simulation of inter- of Raessi et al. [9] . The proposed artificial viscosity model can ac- facial flow applications and has been commonly attributed to the commodate arbitrary interface viscosities and two methods to dy- explicit numerical implementation of surface tension. As a result, namically compute the interface viscosity based on the local flow it is widely postulated that an implicit implementation of surface conditions are presented. We present and discuss the results for tension would lift or at least mitigate the capillary time-step con- a range of numerical experiments, which allow a comprehensive straint [1,8–10,18] . Recent research efforts inspired by this assump- assessment of the efficacy of the methodology, highlight the act- tion aimed at finding an implicit or semi-implicit treatment of the ing physical mechanisms and provide best practice guidelines for surface tension to lift the time-step restrictions in interfacial flows future interfacial flow simulations using the proposed artificial vis- [8,9,11,12] . Hysing [8] proposed a semi-implicit formulation of sur- cosity model. Furthermore, our study demonstrates that the suc- face tension based on the CSF method [1] for a two-dimensional cess of the proposed methodology with regards to mitigating the finite element method. The semi-implicit formulation of Hysing in- capillary time-step constraint is solely based on the dissipation of cludes an additional implicit term which represents artificial shear surface energy, irrespective of the type of implementation, contrary stresses tangential to the interface. Raessi et al. [9] translated this to previous suggestions [8,9] . F. Denner et al. / Computers and Fluids 143 (2017) 59–72 61

The remainder of this article is organised as fol- is discretised using a specifically constructed momentum-weighted lows. Section 2 introduces the governing equations and interpolation method proposed by Denner and van Wachem [6] , Section 3 presents the applied numerical framework. The ar- that couples pressure and velocity and assures a discrete balance tificial viscosity model is presented and appropriate choices of the between pressure gradient, gravity and surface force. interface viscosity as well as the acting physical mechanisms are The solution procedure follows a coupled, implicit implementa- discussed in Section 4 . The results for a variety of numerical test tion of the primitive variables, following the work of Denner and cases are presented in Section 5 , in order to scrutinise the pro- van Wachem [6] , where the governing equations of the flow are posed methodology, show its efficacy and provide a benchmark for solved in a single linear system of equations, given as ⎛ ⎞ ⎛ ⎞ future modelling efforts. The article is summarised and concluded x -momentum equation φ in Section 6 . u ⎜ y -momentum equation⎟ ⎜ φ ⎟ ⎝ ⎠ · ⎝ v ⎠ = b, (10) z-momentum equation φ w 2. Governing equations φ continuity equation p The considered isothermal, incompressible flow of Newtonian A φ fluids is governed by the continuity equation and the momentum where φ is the solution vector, constituted by the solution subvec- equations, defined as tors of velocity u = (u, v , w ) T and pressure p . Each time-step con- ∂u sists of a finite number of non-linear iterations to account for the i = 0 (3) non-linearity of the governing equations. At the end of each non- ∂x i linear iteration the deferred terms of the equation system are up- dated, based on the result of the most recent non-linear iteration. ∂ ∂u i ∂u i ∂ p ∂ ∂u i u j ρ + u = − + μ + + ρ g + f , This iterative procedure continues until the non-linear problem has ∂ j ∂ ∂ ∂ ∂ ∂ i s i t x j xi x j x j xi converged to a sufficiently small tolerance. A BiCGSTAB method in- (4) corporated in the freely-available PETSc library [41,42] is utilised to solve the preconditioned linear equation system. where u is the velocity, p stands for the pressure, ρ is the density, In this study a compressive VOF methodology [43] is applied to μ is the viscosity of the fluid, t represents time, g is the gravi- discretise the temporal evolution of the interface, governed by Eq. tational acceleration and f is the volumetric force due to surface s (9) , using algebraic discretisation schemes to discretise Eq. (9) and tension. Based on hydrodynamic principles, the forces acting on transporting the colour function in a time-marching fashion. This the interface must balance in both phases. Neglecting gradients in compressive VOF methodology inherently conserves mass within surface tension coefficient, the force-balance in the direction nor- the limits of the solver tolerance [43] and is able to capture evolv- mal to the interface is [39] ing interfaces with similar accuracy as VOF-based interface recon- ∂( ) ∂( ) struction methods [43,44] . The interface advection and the flow u j mˆ j u j mˆ j p − p a − 2 μ mˆ − 2 μa mˆ = σκ. (5) b b ∂ i ∂ i solver are implemented in a segregated fashion and coupled ex- xi xi b a plicitly, see Ref. [43] for details. where subscripts a and b denote the two fluids, σ is the surface Based on the colour function distribution, the surface force per tension coefficient, κ is the interface curvature and mˆ is the unit unit volume is discretised using the CSF model [1] as normal vector of the interface (pointing into fluid b). ∂γ

The Volume of Fluid (VOF) method [24] is adopted to capture f s ,i = σκmˆ i δ = σκ , (11) ∂x the interface between two immiscible fluids. In the VOF method i the local volume fraction in each cell is represented by the colour where mˆ = ∇ γ / |∇ γ | is the interface normal vector. In order to en- function γ , defined as sure a balanced-force discretisation of the surface force, Eq. (11) is discretised on the same computational stencil as the pressure gra- 0 fluid a γ ( x, t) = (6) dient [6] and no convolution is applied to smooth the surface force . 1 fluid b [45] . The interface is, therefore, situated in every mesh cell with a 4. Artificial viscosity model for interfacial flows colour function value of 0 < γ < 1. The local density ρ and viscosity μ are defined based on the colour function γ , given as The assumption that an implicit treatment of surface tension ρ( , ) = ρ − γ ( , ) + ρ γ ( , ) x t a [1 x t ] b x t (7) can mitigate the capillary time-step constraint, first proposed by Brackbill et al. [1] , has motivated recent efforts to formulate a (semi-)implicit implementation of the CSF model. Following the μ( , ) = μ − γ ( , ) + μ γ ( , ) , x t a [1 x t ] b x t (8) work of Bänsch [46], Hysing [8] proposed an additional implicit term for finite element methods that is dependent on the surface where subscripts a and b denote the two fluids. The advection of tension, which Raessi et al. [9] extended to finite volume methods, the colour function γ is defined based on the underlying flow field given as as

= σ δ ∇ 2 , ∂γ ∂γ fτ,i t s ui (12) + = . ui 0 (9) ∇ 2 ∂t ∂x i where s is the Laplace-Beltrami operator tangential to the interface and δ = |∇γ | is the so-called interface density. The coeffi- 3. Numerical methodology cient σ t δ of the tangential Laplacian of the velocity has the same unit (Pa s) as the dynamic viscosity μ and can, therefore, be The applied numerical framework is based on a finite volume seen as constituting an artificial interface viscosity. Consequently, method with a collocated variable arrangement [6,40] . The tran- the additional term given by Eq. (12) represents artificial shear sient term of the momentum equations, Eq. (4) , is discretised using stresses acting tangential to the interface in the interface region the Second-Order Backward Euler scheme and convection is discre- in addition to the shear stresses imposed by the bulk phases, in- tised using a central differencing scheme. The continuity equation creasing the dissipation of interface movement. 62 F. Denner et al. / Computers and Fluids 143 (2017) 59–72

4.1. Artificial interfacial shear stresses density is 0 ≤ δˆ ≤ 0 . 5 and, hence, the applied interface viscosity ∗ in any given mesh cell is μ ≤ 0 . 5 μ. The base value of the inter- Following the analysis of Eq. (12) in the previous paragraph, the face viscosity can either be fixed to a predefined value or can be additional shear stresses acting tangential to the interface can be calculated based on the fluid properties and the flow field, such as generalised as the two specific examples discussed below. 2 The original formulation of the additional interfacial shear f τ,i = μ ∇ u i , (13) s stress term proposed by Raessi et al. [9] is retained by defining where μ is the interface viscosity, representing a numerical dif- the base value of the interface viscosity as fusion coefficient applied to smooth the topology of the fluid inter- ∗ σ t face. For an isothermal, incompressible flow the momentum equa- μ = . (16) x tions, Eq. (4) , including the additional interfacial shear stress term of the artificial viscosity model become This interface viscosity is proportional to the surface tension coefficient σ and the time-step t but inversely proportional to the ∂ ∂ ∂ ∂ ∂ ∂ ui ui p ui u j mesh spacing x , similar to the magnitude of parasitic currents ρ + u j = − + μ + ∂t ∂x j ∂x i ∂x i ∂x j ∂x i [4,6] . Denner and van Wachem [13] demonstrated that the viscous ∂γ + ρ + σκ + μ ∇ 2 . gi s ui (14) attenuation of interface waves in numerical simulations is criti- ∂x i cally influenced by the mesh resolution, concluding that the at- The prevailing effect of this additional interfacial shear stress term, tenuation of spurious capillary waves cannot be represented in a Eq. (13) , is an increased dissipation of surface energy through a physically accurate manner, since the vorticity generated by these local increase of the acting shear stresses. The underlying mech- waves is typically not discretely resolved by the computational anism closely resembles the additional shear stresses imposed at mesh. Thus, an alternative definition of the interface viscosity is the interface by the adsorption of surface-active substances, which proposed based on the length scale of the penetration depth of the in the literature is often described using a surface viscosity term, vorticity generated by capillary waves, given as [49] following the work of Scriven [47] , that is identical to Eq. (13) . ν With respect to surface-active substances the surface viscosity it- l ν = , (17) self is, strictly speaking, not a fluid or interface property but should ω σ merely be regarded as a phenomenological constant [39] . The ad- where ν = μ/ρ is the kinematic viscosity. With respect to the dis- ditional interfacial shear stress term, Eq. (13) , also bears similarity persion of capillary waves, it can be assumed that the fluid prop- to other types of artificial viscosity models, such as the original ar- erties of both phases act collectively on capillary waves [48] , so tificial viscosity model of von Neumann and Richtmyer [32] or the that for instance the effective kinematic viscosity is ν = νa + ν . spectral-like artificial viscosity model of Cook and Cabot [35] . b Following this rational for the purpose of defining a length scale With respect to parasitic currents, the additional interfacial that takes into account the natural attenuation of capillary waves, shear stresses play the traditional role of increasing the dissipa- for fluids with different fluid properties and including interface vis- tion of parasitic currents, in particular since parasitic currents are cosity, this viscous length scale can be reformulated as spatially very localised and the associated second spatial deriva- μ μ μ tives are typically high. The effect of the additional interfacial 1 a b l ν = + + , (18) shear stresses on spurious capillary waves is more versatile, as ω ρ ρ ρ σ a b they reduce the frequency and phase velocity of capillary waves ρ = (ρ + ρ ) / [48] and also increase the penetration depth of the vorticity in- where a b 2 is the reference density at the interface. duced by capillary waves [49] , leading to (increased) attenuation Hence, the base value of the interface viscosity follows as μ μ of these waves. Furthermore, the increased viscosity acting on cap- ∗ 2 a b μ = max ρ l ν ω σ − − , 0 . (19) ρ ρ illary waves steepens the energy cascade of capillary wave turbu- a b lence [50] and, thus, capillary waves with small wavelength that This formulation of the interface viscosity takes into account the result from wave interaction are less energetic. damping provided by the bulk phases as well as the increased attenuation of interface features with decreasing mesh spacing, since μ 4.2. Choice of interface viscosity flow structures associated with parasitic currents and spurious capillary waves equally reduce in size. These flow structures ex-

Defining an appropriate interface viscosity for a given problem perience a larger natural attenuation because viscosity acts prefer- is essential for the efficacy of the artificial viscosity model and the ably at smaller scales [23] . Furthermore, since underresolved in- overall accuracy of the conducted simulation. If the interface vis- terface waves are underdamped due to an inadequate resolution cosity is too small, parasitic and spurious artefacts cannot be coun- of the vorticity induced by these waves [13] , a reduced mesh spac- teracted with maximum effect. On the other hand, if the interface ing ( i.e. higher mesh resolution) increases the viscous dissipation of viscosity is too large, the artificial viscosity model compromises those waves. If the bulk phases provide sufficient damping, e.g. in the predictive quality of the numerical framework and changes the the case of overdamped capillary waves [48] , the interface viscosity outcome of the simulation. As a general template it is proposed to becomes zero. Following the argument of Prosperetti [49] , the vis- define the interface viscosity as cous length scale in Eq. (19) is defined as l ν = x . Thus, the result- ∗ ing interface viscosity leads to a penetration depth of the vorticity μ = μ δˆ (15) generated by an interfacial wave that is of the order of magnitude ∗ where μ is the base value of the interface viscosity, δˆ = δ x of the mesh spacing. The angular frequency ω σ should be based is the normalised interface density and x is the mesh spacing. on a representative wavelength, such as the minimum spatially re- λ = This formulation assures that the interface viscosity is non-zero solved wavelength min 2 x or the smallest wavelength that is λ = λ = only in the interface region as well as that the shear stress term spatially adequately resolved, e.g. 3 min 6 x. Note that this of the artificial viscosity model is distributed in the same way as represents only one particular choice for the viscous length scale l ν the force due to surface tension and, hence, only acts at the inter- and the angular frequency ω σ for Eq. (19) and that other choices face. As shown schmematically in Fig. 1 , the normalised interface may be more suitable for a given problem. F. Denner et al. / Computers and Fluids 143 (2017) 59–72 63

ˆ Fig. 1. One-dimensional examples of an equidistant Cartesian mesh with colour function γ and corresponding normalised interface density δ = |∇γ | x, with |∇γ j | ≈ | γ j+1 − γ j−1 | / (2 x ) discretised using central differencing.

The particular choice of interface viscosity also dominates the since t μ∝ x 2 and t σ ∝ x 3/2. Comparing t μ and t σ , the crit- spatiotemporal convergence behaviour of the artificial viscosity ical mesh spacing follows as model. The interface viscosity defined in Eq. (16) is O(x −1 , t) . μ2 2 Thus, the interface viscosity diverges with first-order under mesh x = , (24) ∗ c πσρ refinement, with μ = ∞ for x → 0, and converges with first- μ∗ = → order for decreasing time-steps, with 0 for t 0. The in- and the largest stable time-step becomes terface viscosity defined in Eq. (19) with l ν = x, on the other ∗ > hand, converges with order 0.5 for spatial refinement, since μ ∝ tσ if x xc √ t = (25) max < . x, and is independent of the time-step t . tμ if x xc

However, in many cases of practical interest x x c with t σ 4.3. Implementation t μ and, hence, the viscous time-step constraint does not impose an additional limit to the applied time-step for such cases. Hysing [8] and Raessi et al. [9] implemented Eq. (12) implic- The magnitude of capillary and viscous time-step constraints is fur- itly, assuming that an implicit implementation is required to lift ther discussed for each individual test case in Section 5 . An implicit ∇ 2 ∇ the capillary time-step constraint. This widely advocated assump- implementation of s u and s u [see Eqs. (21) and (22), respec- tion has been corrected by Denner and van Wachem [13] , demon- tively], as for instance described in detail by Raessi et al. [9] , can strating that restrictions imposed by the capillary time-step con- remedy this issue for cases in which t μ t σ . straint also hold for the implicit implementation of surface tension. Contrary to previous studies [8,9] , the additional interfacial 5. Results shear stress term of the artificial viscosity model, Eq. (13) , is implemented as an explicit contribution to the momentum equations, The results of four representative test-cases are presented, scru- defined as tinising the proposed artificial viscosity model for interfacial flows − by highlighting its acting mechanisms and discussing its impact j = μ ∇ 2 j 1 , fτ,i s ui (20) ∗,t on realistic applications. In what follows, μ denotes the inter- where superscripts j and j − 1 denote the current and the previous face viscosity based on the work of Hysing [8] and Raessi et al. non-linear iteration, respectively. The Laplace-Beltrami operator of [9] , see Eq. (16) , with superscript t indicating its time-step depen- ∗,λ μ velocity with respect to the tangential vector of the interface is dency, and denotes the interface viscosity determined by Eq. (19) , where superscript λ stands for its dependency on the chosen

∇ 2 = ∇ · ∇ − ( · ∇)(∇ ) , s ui s ui mˆ s ui mˆ (21) reference wavelength. For all presented simulations, the applied time-step satisfies the capillary time-step constraint, Eq. (2) , the where viscous time-step constraint, Eq. (23) , as well as a Courant number Co = | u| t/ x < 1 , unless explicitly stated otherwise. ∇ s u i = ∇u i − ( mˆ · ∇u i ) mˆ (22) is the velocity gradient tangential to the interface. 5.1. Dispersion of capillary waves Because the shear stress term of the artificial viscosity model is implemented explicitly, the applied time-step has to fulfil the A sinusoidal, surface-tension-driven wave between two viscous viscous time-step constraint [9] fluids is simulated. The wave with wavelength λ has an initial am- 2 plitude of 0.01 λ and the Laplace number of the two-fluid system is ρ x t μ ≤ . (23) La = σρλ/μ2 = 30 0 0 . Both fluids are initially at rest, have a den- 2 μ − − sity of ρ = 1 . 0 kg m 3, a viscosity of μ = 1 . 6394 × 10 3 Pa s and a − − Since the Cauchy stress tensor of the momentum equations, Eq. (4) , surface tension coefficient σ = 0 . 25 π 3 N m 1, which results in a is implemented implicity in the applied numerical framework, only non-dimensional viscosity of the interface viscosity μ of the artificial viscosity model has to 2 4 μπ − be considered in determining the viscous time-step constraint by = = 6 . 472 × 10 2. (26) ρω σ λ2 Eq. (23) . However, strict adherence to the viscous time-step constraint is in many cases not necessary, since the solution algorithm The motion of the interface is induced by surface tension only, accounts iteratively for non-linearities of the governing equations gravity is excluded. The domain is λ in width ( x -direction) and 3 λ and, hence, the additional interfacial shear stress term, Eq. (20) , in height ( y -direction), identical to the domain used by Popinet is changing in every iteration. Preliminary tests have shown that, [18] , and is represented by an equidistant Cartesian mesh with with the numerical framework presented in Section 3 , stability is- a spatial resolution of 32 cells per wavelength λ. The periodic sues associated with the viscous time-step constraint only arise for motion of the capillary wave is discretised with t = 10 −3 s for very large interface viscosities. λ = 1m and t = 3 . 162 × 10 −5 s for λ = 0 . 1m , which corresponds − Dependent on the mesh spacing, the fluid properties and the to 10 0 0 time-steps t per period ω σ 1. All domain boundaries applied interface viscosity, the viscous time-step constraint t μ are treated as free-slip walls. The capillary time-step constraint can be more restrictive than the capillary time-step constraint t σ , is t σ = 3 . 471 × 10 −2 s for the wave with λ = 1 m and t σ = 64 F. Denner et al. / Computers and Fluids 143 (2017) 59–72

∗ Fig. 2. Temporal evolution of the amplitude A of the standing capillary wave with wavelength λ = 1 m and λ = 0 . 1 m for different interface viscosities μ . The analytical result is based on the solution of the initial-value problem of Prosperetti [51] .

The temporal evolution of the capillary wave amplitude A , with and without interface viscosity is shown in Fig. 4 for different ∗ mesh resolutions. Without interface viscosity ( μ = 0 ), see Fig. 4 a, the prediction of the wave amplitude becomes more accurate, compared to the analytical results, with increasing mesh resolution. The results obtained on meshes with 64 cells and 128 cells per wavelength λ are almost indistinguishable, indicating that a resolution of 64 cells per wavelength is sufficient. Applying the ∗ −2 artificial viscosity model with μ = 10 Pa s , see Fig. 4 b, the influence of the artificial viscosity model is diminishing with increasing mesh resolution. Since the artificial viscosity model is only applied in the close vicinity of the interface, the additional shear stresses introduced by the artificial viscosity model are acting in a region of decreasing volume for meshes with increasing resolution, thereby decreasing the added dissipation. The capillary ζ = /ω Fig. 3. Damping ratio 0 of the standing capillary wave with wavelength λ = λ = . μ∗ and viscous time-step constraint for the case with 128 mesh cells 1 m and 0 1 m for different interface viscosities . ∗ −2 −3 per wavelength and μ = 10 Pa s are t σ = 4 . 339 × 10 s and

t μ = 1 . 221 × 10 −2 s , respectively.

−3 ∗ −2 1 . 098 × 10 s for λ = 0 . 1m . Assuming μ = 10 Pa s , the vis- 5.2. Interface instabilities on falling liquid films cous time-step constraint is t μ = 1 . 953 × 10 −2 s for λ = 1 m and 1 . 953 × 10 −3 s for λ = 0 . 1 m . Prosperetti [51] derived an analyti- Falling liquid films are convectively unstable to long-wave per- cal solution for the initial-value problem of the evolution of such turbations ( i.e. the wavelength is much longer than the film height) a capillary wave in the limit of small wave amplitude and equal which leads to the formation of periodic or quasi-periodic wave kinematic viscosity of the bulk phases. trains [52] . Waves with a frequency higher than the neutral sta- Fig. 2 shows the temporal evolution of the amplitude for wave- bility frequency f are attenuated by surface tension, whereas lengths λ = 1 m and λ = 0 . 1 m for the predefined interface viscosi- crit waves with a frequency lower than f evolve as a result of the ties μ∗ = 0 Pa s ( i.e. no artificial viscous shear stresses are applied crit long-wave instability mechanism [53] . The resulting solitary waves at the interface), μ∗ = 10 −3 Pa s and μ∗ = 10 −2 Pa s . The simula- are governed by complex hydrodynamic phenomena and exhibit tion results for both cases without artificial viscosity model ( μ∗ = a dominant elevation with a long tail and steep front, typically 0 ) are in excellent agreement with the analytical solution of Pros- with capillary ripples preceding the main wave hump. In inertia- peretti [51] , which is shown in the graphs as a reference. The tem- dominated film flows, solitary waves exhibit a separation of scales poral evolution of the amplitude of the capillary wave deviates in- between the front of the main wave hump, where gravity, viscous creasingly from the analytical solution as the interface viscosity in- drag and surface tension balance, and the tail of the wave, charac- creases, due to the enhanced attenuation through the additional terised by a balance between gravity, viscous drag and inertia [52] . shear stresses imposed by the artificial viscosity model, given by Two different film flows are considered to study the effect of the damping coefficient the artificial viscosity model: a) the attenuation of numerical arte-

ln (| A | / | A | ) facts at the interface, and b) the spatiotemporal aliasing of spuri- = 0 1 , (27) ous capillary waves. Both cases are simulated in a domain of size t 1 − t 0 × × Lx Ly 0.1hN , schematically illustrated in Fig. 5, represented by where A is the wave amplitude, t stands for time and superscripts an equidistant Cartesian mesh with 10 cells per ﬁlm height hN ,

0 and 1 denote two extrema with respect to the temporal evo- where hN is the height of the unperturbed film (Nusselt height) lution of the wave amplitude. Fig. 3 shows the damping ratio for a given flow rate according to the Nusselt flat film solution ζ = /ω σ of both waves as a function of interface viscosity. The [52] . For all cases considered in this study the domain height is damping ratio increases rapidly with increasing interface viscos- L y = 4 h N . The inclined substrate has an inclination angle β to the ity and, as previously indicated, the damping effect of the artifi- horizontal plane and is modelled as a no-slip boundary. The gas- cial viscosity model on the shorter wave ( λ = 0 . 1 m ) is significantly side (top) boundary is assumed a free-slip wall. A monochromatic stronger. perturbation is imposed at the domain-inlet, periodically changing F. Denner et al. / Computers and Fluids 143 (2017) 59–72 65

∗ ∗ −2 Fig. 4. Temporal evolution of the amplitude A of the standing capillary wave with wavelength λ = 1 m and interface viscosities μ = 0 and μ = 10 Pa s for different mesh resolutions, indicated by the number of cells per wavelength λ. The analytical result is based on the solution of the initial-value problem of Prosperetti [51] .

for this case is t σ = 9 . 927 × 10 −5 s , whereas the viscous time- −4 ∗ ∗,t step constraint is t μ = 9 . 965 × 10 s for μ = μ and t μ = −4 ∗ ∗,λ 1 . 163 × 10 s for μ = μ . Fig. 6 a shows the ﬁlm height h , normalised by the Nusselt ﬂat

film height hN , and Fig. 6b shows the streamwise velocity at the , interface u normalised by the Nusselt velocity uN , as a function of the normalised downstream distance x . Experimental results for this particular case of Denner et al. [56] are shown as a reference. The impact of numerical artefacts at the front of the solitary wave Fig. 5. Schematic illustration of the numerical domain with dimensions L × L . The x y liquid film with height h ( x, t ) flows from left to right on substrate with inclination is clearly visible in Fig. 6, manifesting as wiggles of the film height angle β. h (see inset of Fig. 6 a), particularly between the wave crest and the preceding trough, and as strong fluctuations of the interface velocity u (see inset of Fig. 6 b). Since the applied time-step sat- the mass flow at frequency f and amplitude A from the mean, with isfies t σ , no external perturbations of high frequency are applied a semi-parabolic velocity profile prescribed for the liquid phase at and given that the numerical artefacts occur in a region of rela- the inlet tively high curvature and low shear stress, the observed numerical artefacts are most likely parasitic currents. No such wiggles 3 y y 2 ( = , ≤ ≤ ) = + ( π ) − , u x 0 0 y hN [1 A sin 2 f t ] 2 uN and fluctuations are observed if the artificial viscosity model is ap- 2 h 2 N hN plied with either of the two considered definitions of the interface (28) viscosity. However, the artificial viscosity model noticeably influences the hydrodynamics of the capillary ripple preceding the soli- and a spatially-invariant velocity prescribed at the inlet for the ∗,t tary wave. With increasing interface viscosity ( μ = 0 . 0334 Pa s gas-phase ∗,λ and μ = 0 . 277 Pa s ) the amplitude of the capillary ripple preced- 3 ing the solitary wave is considerably reduced and its wavelength is u (x = 0 , h < y ≤ 4 h ) = [1 + A sin ( 2 π f t )] u , (29) N N 2 N increased, leading to smaller variations in interface velocity. Capil- lary ripples form in order for the surface energy of the interface to where uN is mean flow velocity (Nusselt velocity) based on the Nusselt flat film solution [52] , given as balance the inertia of the flow and an increasing number of capillary ripples is generally observed for increasing inertia [52,58] . As ρ β 2 l g sin hN a consequence of the artificial viscosity model increasing the dissi- u = . (30) N μ 3 l pation in the vicinity of the interface (which is acting particularly at short wavelengths [48] ) and, therefore, dissipating some of the The film height h (x = 0) = h N at the inlet is constant and an open inertia of the flow, the resulting capillary ripple is smaller for in- boundary condition is applied at the domain-outlet [54,55] . The creasing interface viscosity. film is initially flat and the velocity field is fully developed.

5.2.1. Mitigating parasitic currents 5.2.2. Spatiotemporal aliasing of spurious capillary waves Following the work of Denner et al. [56] and Charogiannis et al. Since the interface and the flow field are clearly oriented, falling [57] , the considered liquid film has a Reynolds number of Re = liquid films are well suited to study the onset of spatiotemporal ρ /μ = . l uN hN l 20 55 and is flowing down a substrate with an in- aliasing of spurious capillary waves as a result of breaching the clination angle of β = 20 ◦ to the horizontal. The liquid phase has capillary time-step constraint given in Eq. (2) , as demonstrated by ρ = . −3 μ = . × a density of l 1169 165 kg m and a viscosity of l 1 8433 Denner and van Wachem [13]. Spurious capillary waves have typ- −2 −3 10 Pa s , the gas phase has a density of ρg = 1 . 205 kg m and ically a frequency that is significantly larger than the neutral sta- μ = . × −5 , a viscosity of g 1 82 10 Pa s and the surface tension co- bility frequency fcrit and, hence, from a purely physical viewpoint, efficient is σ = 5 . 8729 × 10 −2 N m −1 . The Nusselt height for this spurious capillary waves should be naturally attenuated. Raessi −3 film flow is h N = 1 . 65923 × 10 m and the Nusselt velocity is u N = et al. [9] reported that an implicit implementation of the addi- 1 . 95296 × 10 −1 m s −1 . The length of the applied three-dimensional tional interfacial shear stress term, Eq. (12) , allowed them to con- −1 domain is L x = 205 h N and the perturbation frequency is f = 7 s duct numerically stable simulation with time-steps up to 5 times with an amplitude of A = 11 . 38% . The capillary time-step constraint larger than the capillary time-step constraint of Brackbill et al. [1] . 66 F. Denner et al. / Computers and Fluids 143 (2017) 59–72

Fig. 6. Film height h , normalised with the Nusselt ﬁlm height h N , and streamwise velocity at the interface u , normalised with the Nusselt velocity u N , as a function of normalised downstream distance x . The insets show a magniﬁed view of the front of the main wave hump, which is visibly affected by parasitic currents. The corresponding ∗,t −2 ∗,λ −1 reference interface viscosities are μ = 3 . 34 × 10 Pa s and μ = 2 . 77 × 10 Pa s . Experimental results of Denner et al. [56] are shown as a reference.

As previously discussed in the introduction of this article, Hys- ing [8] as well as Raessi et al. [9] attributed this behaviour to the implicit implementation of this term, a hypothesis which has later been disputed by Denner and van Wachem [13] . In the current study the additional shear stress term of the artificial viscosity model, Eq. (13) , is implemented explicitly, as detailed in Section 4.3 . A falling water film on a vertical ( β = 90 ◦) substrate is simulated, as previously considered experimentally and numerically by Nosoko and co-workers [54,59] . The liquid phase has a den- ρ = −3 μ = −3 , sity of l 998 kg m and a viscosity of l 10 Pa s whereas −3 the gas phase is taken to have a density of ρg = 1 . 205 kg m −5 and a viscosity of μg = 10 Pa s . The surface tension coefficient is σ = 7 . 2205 × 10 −2 N m −1 . The liquid film flow has a Reynolds −4 number of Re = 51 . 1 with h N = 2 . 51 × 10 m . The computational domain has a length of L x = 300 h N and the perturbation frequency is f = 25 s −1 with an amplitude of A = 3% . Given the neutral stabil- = . −1 , ity frequency for this case is fcrit 169 3 s the shortest spatially λ = (λ ) = . × resolved capillary waves with min 2 x and f min 59 9 10 3 s −1 are naturally attenuated and, hence, any observed growth of these waves is a numerical artefact. Fig. 7 a and b shows the film height as a function of downstream distance of the falling water film at two time instants for different numerical time-steps. The capillary time-step constraint −6 following Eq. (2) is t σ = 5 . 52 × 10 s , assuming u · k = 1 . 5 u N (which is the streamwise interface velocity of the unperturbed film). In Fig. 7 a the onset of spatial aliasing for the case with −6 ∗ t = 8 . 31 × 10 s and μ = 0 is starting to be visible, even in the ࣡ flat section of the film for x 160 hN , while the other two cases do not exhibit any aliasing. The aliasing severely influences the evolution of the falling liquid film shortly after its onset, see Fig. 7 b, in particular at the crest and the preceding trough of the largest interfacial wave. No aliasing is observed if the time-step is reduced to satisfy the capillary time-step constraint ( t < t σ ), exempli-

fied by the case with t = 3 . 92 × 10 −6 s ≈ 0 . 71 t σ in Fig. 7 . Ap- ∗ ∗,t plying the artificial viscosity model with μ = μ = 0 . 0239 Pa s , Fig. 7. Film height h , normalised with the Nusselt film height h , as a function of for which t μ = 2 . 634 × 10 −5 s applies, increases the dissipation of N normalised downstream distance x of the vertically falling water film with pertur- = − the surface energy of spurious capillary waves and, even with t bation frequency f = 25 s 1 at (a) time t = 0 . 065 s and (b) t = 0 . 075 s for different − 8 . 31 × 10 6 s ≈ 1 . 5 t σ , results in a smooth and accurate evolution time-steps t . The insets show magnified views of the film height at specific down- μ∗,t = . of the interface without aliasing. Note that the interface viscosity stream distances. The reference interface viscosity is 0 0239 Pa s and the cap- = . × −6 is 23.9 times larger than the viscosity of the liquid and is, there- illary time-step constraint is tσ 5 52 10 s. fore, increasing the penetration depth of the vorticity generated by capillary waves, see Eq. (17) , by almost factor 5. This leads to a significantly increased attenuation of spurious capillary waves, which viscosity and would, therefore, have a more pronounced impact on is essential for the ability to breach the capillary time-step con- the accuracy of the simulation results. It is worth noting that de- straint t σ . It is possible to apply a larger time-step than in the spite the explicit implementation of the artificial viscosity model, shown example but this would generally require a larger interface the numerical algorithm is stable and spurious capillary waves are F. Denner et al. / Computers and Fluids 143 (2017) 59–72 67

= ρ 2 / = Fig. 8. Temporal evolution of (a) the Froude number Fr and (b) the total kinetic energy Ekin u 2 in the computational domain of the rising bubble with Eo 40 and Mo = 0 . 056 .

Fig. 9. Capillary instability with wavelength λ and amplitude η on a liquid jet with radius r and initial radius r 0 .

direction. Both ﬂuids are initially at rest and the motion of the bubble is induced by buoyancy only. The applied computational × × domain has a size of 5d0 7d0 5d0 and is resolved with an equidistant Cartesian mesh of 100 × 140 × 100 cells. The boundary at the top of the domain is considered to be an outlet boundary, all other boundaries are free-slip walls. Given the domain width of

5d0 , the rise velocity in the applied computational domain is expected to be 96% of the rise velocity observed in a domain of infinite extend [6] . The capillary time-step constraint is t σ = 9 . 185 × − 10 −4 s and the viscous time-step constraint is t μ = 1 . 143 × 10 2 s ∗ ∗,t −3 ∗ ∗,λ for μ = μ and t μ = 1 . 660 × 10 s for μ = μ . Empirical studies by Clift et al. [61, Fig 2.5] suggest a terminal = | | ρ /μ ≈ . − . Reynolds number of Red u r o d0 o 20 5 21 0 for a bub- = = . ble with Eod 40 and Mo 0 056 in a domain of infinite ex- = | | / ≈ tend, which represents a Froude number of Frd ur d0 g 0 . 626 − 0 . 642 with respect to the chosen fluid properties and ≈ . − . Frd 0 601 0 616 considering the finite width of the applied computational domain. Denner et al. [44] reported a terminal Fig. 10. Comparison of the numerical breakup length Lb and the theoretical = . breakup length Lb, R based on the linear stability analysis of Rayleigh [63] for differ- Froude number of Frd 0 58 using a VOF-PLIC method in a do- μ∗ ent interface viscosities and for different time-steps t. main of 5d0 width and Pivello et al. [60] obtained a terminal = . Froude number Frd 0 606 using a front-tracking method and a

domain of 8d0 width. not ampliﬁed, allowing to breach the capillary time-step constraint Fig. 8 a shows the temporal evolution of the Froude number ∗, without consequences. μ t = . , Frd . The additional viscous dissipation imposed by 0 07 Pa s Eq. (16) , causes only a very small difference in the Froude num-

5.3. Buoyancy-driven rise of a bubble ber ( i.e. rise velocity) of the bubble compared to the case with- ∗,λ out interface viscosity. Calculating the interface viscosity μ =

The rise of a bubble due to the sole action of buoyancy, char- 0 . 6085 Pa s based on Eq. (19) , on the other hand, results in a no- = μ4 /ρ σ 3 = . acterised by its Morton number Mo g o o 0 056 and its ticeably affected Froude number, which is particularly visible in the Eötvös number Eo = ρ g d 2/σ = 40 is simulated and analysed. d o 0 magnification of Fig. 8 a. Despite the observed differences as a re- ρ = −3 The continuous phase has a density of o 1000 kg m and a vis- sult of the artificial viscosity model and corresponding interface μ = . cosity of o 0 2736 Pa s. Following previous studies [6,43,44,60], viscosity, the Froude numbers for all three cases are well within ρ /ρ = μ /μ = the density and viscosity ratio of the bubble are i o i o the expected range of Froude numbers as given by empirical stud- −2 = 10 . The bubble is initially spherical with a diameter of d0 ies and numerical results discussed in the previous paragraph. The . , σ = . −1 0 02 m the surface tension coefficient is 0 1 N m and the total kinetic energy Ekin of the flow in the entire computational do- gravitational acceleration of g = 10 m s −2 is acting in negative y - 68 F. Denner et al. / Computers and Fluids 143 (2017) 59–72

Table 1 ∗ ∗ −6 Values of μ and μ /μ for different simulation time-steps t . Based on Eq. (2) , t σ 3 . 4 × 10 s .

Fig. 11 a 12 a 13 a 11 b 12 b 13 b 11 c 12 c 13 c 11 d 12 d 13 d

t (×10 −6 s) 0.5 1.0 3.4 6.8 t / t σ 0.15 0.29 1.0 2.0 ∗ −3 μ (×10 Pa s) 0.0 1.46 2.92 0.0 2.92 5.84 0.0 9.93 19.9 0.0 19.9 39 .8 ∗ μ /μ 0.0 1.62 3.24 0.0 3.24 6.48 0.0 11.0 22.1 0.0 22.1 44 .2 − t μ (×10 6 s) ∞ 463.2 213 .6 ∞ 213 .6 106.3 ∞ 56.72 28.22 ∞ 28.22 14 .11

−3 ∗ −6 Fig. 11. Pulsed water jet at t = 3 . 06 × 10 s with μ = 0 and for t ∈ { 0 . 5 , 1 . 0 , 3 . 4 , 6 . 8 } × 10 s .

−3 −5 main exhibits a decreasing total kinetic energy in the domain for sity of ρg = 1 . 18 kg m and a viscosity of μg = 1 . 85 × 10 Pa s . increasing interface viscosity as a results of the increased viscous The liquid jet has a diameter of d o = 2 r o = 0 . 25 mm and axial ve- −1 dissipation, as seen in Fig. 8b. locity U o = 3 . 0 m s , with a surface tension coefficient of σ = 7 . 3 × 10 −2 N m −1 . This corresponds to a Weber number of W e = √ ρ 2 /σ = . = μ/ ρσ = 5.4. Capillary instability of a liquid jet ro Uo 15 365 and an Ohnesorge number of Oh ro 9 . 44 × 10 −3 . A gravitational acceleration of g = 9 . 81 m s −2 is acting The capillary instability of a water jet, schematically illustrated in the positive x -direction. The applied computational domain is of in Fig. 9 , is simulated and analysed in order to assess the influ- size 40 d o × 4 d o × 4 d o and is resolved with an equidistant Carte- ence of the proposed artifical viscosity model on the prediction of sian mesh of 400 × 40 × 40 cells. The liquid jet is initialised as a the breakup length of the jet when the capillary time-step con- cylinder with a diameter d o and velocity U o , the gas phase is ini- traint is breached. Following the properties used by Moallemi et al. tially at rest. Finally, the x0 boundary is considered to be an inlet 2 −3 [62] , the liquid phase has a density of ρ = 9 . 97 × 10 kg m and a boundary, the x1 boundary is considered to be a pressure outlet viscosity of μ = 9 . 0 × 10 −4 Pa s , whereas the gas phase has a den- boundary, and all other boundaries are outlets. F. Denner et al. / Computers and Fluids 143 (2017) 59–72 69

−3 ∗ ∗,t −6 Fig. 12. Pulsed water jet at t = 3 . 06 × 10 s with μ = μ and for t ∈ { 0 . 5 , 1 . 0 , 3 . 4 , 6 . 8 } × 10 s .

Considering the equations of motion for an inviscid fluid in a where I n is the modified Bessel’s function of the n -th order. The cylindrical coordinate system ( r, θ, x ), Rayleigh [63] conducted a dimensionless disturbance growth rate linear stability analysis on the jet radius r under the assumption 3 that r ρ ω = o q σ (34) r(θ, x, t) = r o + αn cos (nθ ) cos (kx ) , (31) reaches its maximum ω max ࣃ 0.3433 for κ ࣃ 0.697 [63] . α 1 with n ro , and showed that the original equilibrium becomes Assuming that the linear approximation remains valid until the unstable for n = 0 and for a reduced wavenumber κ = kr o such jet breaks up, the theoretical breakup time tb is obtained by calcu- that | κ| < 1. In that case, the jet radius can be rewritten as qt lating the time necessary for the radial disturbance = o e de- qt fined in Eq. (32) to reach = r o , which gives r(x, t) = r o + o e cos (kx ) ∀ θ, (32)

ln ( r o /o ) ln ( 1 /ηo ) and the growth rate of the unstable radius disturbance q is given t = = , (35) b q q by

where ηo = o /r o is the dimensionless amplitude of the radial dis- σ κI (κ) 2 1 2 q = (1 − κ ) , (33) turbance. The breakup length Lb follows from the breakup time tb 3 ρ (κ) ro Io as

U o ln ( 1 /ηo ) = = . Lb Uo tb (36) 1 In fact, Plateau [64] ﬁrst proved that a liquid jet is stable for all purely q non-axisymmetric deformations, but unstable to axisymmetric modes with | κ| <

1 [ i.e. modes whose wavelength λ(= 2 π r o /κ) > 2 π r o ]. Rayleigh [63] added to In the cases considered in this study, the inlet jet radius remains Plateau’s theory by describing the dynamics of the instability. constant and equal to r o , while the inlet jet velocity periodically 70 F. Denner et al. / Computers and Fluids 143 (2017) 59–72

−3 ∗ ∗,t −6 Fig. 13. Pulsed water jet at t = 3 . 06 × 10 s with μ = 2 μ and for t ∈ { 0 . 5 , 1 . 0 , 3 . 4 , 6 . 8 } × 10 s .

ξ ∗,t changes with dimensionless amplitude o following μ (or a multiple thereof), given in Eq. (16) , is deemed to be the

= + ξ κ / . most adequate choice for the evaluation of the interface viscosity Uinlet Uo [1 o sin ( Uo t ro )] (37) for the considered case, because of its dependency on the time- Based on the mechanical energy of the perturbation, Moallemi step t . Alternatively, the values for μ∗ given in Table 1 could, η et al. [62] derived a relationship between a radius perturbation o of course, be applied explicitly. Fig. 10 shows the breakup length and a velocity perturbation ξ o , given as Lb normalised by the breakup length obtained from linear stabil- = . , 2 ity analysis, Lb , R 24 2 as a function of the simulation time-step ξo = ηo . (38) ∗ ∗ 3 for three values of μ. The additional dissipation imposed by μ Consequently, by inserting Eq. (38) in Eq. (36) , the breakup length increases the breakup length. For the case with the largest time- −6 ξ step t = 2t σ = 6 . 8 × 10 s and the highest interfacial viscosity for a jet with a velocity perturbation o becomes ∗, μ∗ = μ t = . μ, 2 44 2 the breakup length Lb is approximately 11% U 2 = o . higher than for the same case without interface viscosity. This ob- Lb ln ξ (39) q 3 o servation stands in agreement to previous findings [65] , which sug- The liquid jet in the presented simulations is pulsed by a gest that an increase in viscosity increases the breakup time and monochromatic perturbation imposed at the domain-inlet, period- length. The artificial viscosity model dissipates energy at the in- ically changing the mass flow with reduced wavenumber κ = 0 . 7 terface, thereby reducing the capillary-driven instability. Note that (to trigger the most unstable mode) and amplitude ξo = 0 . 08 from a perfect agreement with the result of the linear stability analysis the mean. Based on the linear stability analysis of Rayleigh [63] , is, of course, not expected, due to the limiting assumptions of the the theoretical breakup length following from Eq. (39) with an ini- linear stability analysis as well as the limitations imposed by the ξ = . = . tial amplitude of o 0 08 is Lb , R 24 2 ro . finite spatiotemporal resolution of the conducted simulations and ∗ ∗,t ∗,t Simulations are conducted for μ ∈ { 0 , μ , 2 μ } using differ- the associated difficulties in predicting what is a singular breakup ∗ ent numerical time-steps t , see Table 1 . The interface viscosity event in reality [66] . However, the presented results with μ = 0 F. Denner et al. / Computers and Fluids 143 (2017) 59–72 71 and t < t σ are in very good agreement with the results re- Following the extensive analysis presented for the dispersion of ported by Moallemi et al. [62] for a water jet with the same fluid capillary waves, the evolution of long-wave instabilities on falling properties and only marginally smaller jet velocity (W e = 14 . 8 and liquid films, the buoyancy-driven rise of a bubble and the capillary- Oh = 9 . 44 × 10 −3 ) as considered here. driven breakup of a liquid jet, it can be concluded that for an inter- Figs. 11 –13 show a slice of the computational domain at t = face viscosity of μ 3 μ the impact on the accuracy of the results 3 . 06 × 10 −3 s with the interface contour and the underlying veloc- is negligible. However, dependent on the particular case, higher in- ity field for each of the considered time-steps and interface vis- terfacial viscosities may be applicable without distorting the result cosities. As previously seen in Fig. 10 , the breakup is delayed by in a significant manner. For instance, in the presented case of the the additional interface viscosity imposed through the artificial vis- long-wave instabilities on falling liquid films the applied interface cosity model. This can also be concluded from comparing Figs. 11 d viscosity μ ≈ 23 μ prevents the onset of aliasing if the capillary ∗ and 13 d. In the case of μ = 0 depicted in Fig. 11 d, breaching the time-step constraint is breached, while the influence of the artifi- capillary time-step limit results in visible oscillations of the in- cial viscosity model on the evolution of the long-wave instability is terface as well as the velocity field. These oscillations are absent insignificant despite the high interfacial viscosity. This can be ex- ∗ if the artificial viscosity model is applied, i.e. μ > 0 , of which plained by the long wave length of the examined interfacial waves the results are shown in Figs. 12 d and 13 d. This is due to the in this case. artificial viscosity model damping the (spurious) capillary waves with the shortest wavelengths. In general, the artificial viscosity Acknowledgements has a negligible influence on the overall accuracy of the solution μ∗ μ, for 3 as for instance observed by comparing the velocity The authors are grateful to the Engineering and Physical Sci- fields in Figs. 11 a-b and 12 a-b, and based on the results plotted in ences Research Council ( EPSRC ) for their financial support through Fig. 10 . grant EP/M021556/1 and to PETROBRAS for their financial support. Data supporting this publication can be obtained from https: 6. Conclusions //doi.org/10.5281/zenodo.166716 under a Creative Commons Attri- bution license. An artificial viscosity model to mitigate the impact of numerical artefacts at fluid interfaces has been presented and validated. 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